Low energy photovoltaic conversion in MIND structures

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(1)Low energy photovoltaic conversion in MIND structures Marek Basta. To cite this version: Marek Basta. Low energy photovoltaic conversion in MIND structures. Other. Université de Strasbourg, 2013. English. �NNT : 2013STRAD019�. �tel-01139503�. HAL Id: tel-01139503 https://tel.archives-ouvertes.fr/tel-01139503 Submitted on 6 Apr 2015. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) 1G¶RUGUH :. eFROH'RFWRUDOH0DWKpPDWLTXHV6FLHQFHVGH O ,QIRUPDWLRQHWGHO ,QJpQLHXU UdS ± INSA ± ENGEES. 7+Ê6( SUpVHQWpHSRXUREWHQLUOHJUDGHGH. 'RFWHXUGHO¶8QLYHUVLWpde Strasbourg Discipline : Physique 6SpFLDOLWp3KRWRYROWDwTXH par. Marek BASTA Low-energy photovoltaic conversion in MIND structures Soutenue publiquement le 5 septembre 2013 Membres du jury 'LUHFWHXUGHWKqVH : M. Zbigniew KUZNICKI, professeur, UFR GH3K\VLTXH8QLYHUVLWpGH6WUDVERXUJ Co-'LUHFWHXUGHWKqVH : M. Jan MISIEWICZ, SURIHVVHXU8QLYHUVLWp Technique de Wroclaw Rapporteur externe : M. Uli LEMMER, professeur, Karlsruhe Institut de Technologie, Karlsruhe Rapporteur externe : M. Laurent BIGUE, professeur, 8QLYHUVLWp Haute Alsace Examinateur : M. Marek GODLEWSKI, professeur, Institut GH3K\VLTXHGHO¶$FDGpPLH3RORQDLVGHV6FLHQFHV Examinateur : M. Patrick MEYRUEIS, professeur, Laboratoire ICube, Strasbourg. ICube. UMR 7357.

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(6) TO MY WIFE AND DAUGHTER.

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(8) This work was conducted in cooperation between University of Strasbourg (Alsace, France) and Wroclaw University of Technology (Wroclaw, Poland). I would like to express my gratitude to the French Embassy in Poland and the Regional Parliament of Lower Silesia Province for financial support. I would also like to express my deepest gratitude to the Director of my thesis Prof. Zbigniew T. KUZNICKI for his supervision, inspiration and encouragement during my research. It has been a privilege for me, to be a student of him. I would like to express my gratitude and sincere appreciation to my co-Director Prof. Jan MISIEWICZ for help and guidance and for monitoring my research. I am greatly honored by the kind acceptance of Prof. Uli LEMMER, Prof. Laurent BIGUE, Prof. Marek GODLEWSKI and Prof. Patrick MEYRUEIS as members of the jury for my thesis and would like to thank them for serving as my advisory committee. Special thanks to Prof. Yoshitate TAKAKURA for his incredible introduction to rigorous treatment of electromagnetic propagation and Dr. Sylvain LECLER for his support with numerical calculations. I would like to extend my gratitude to Dr. Victorien RAULOT for the structural measurements and Dr. Bartlomiej WITKOWSKI for the electron microscopy measurements. I would like to give special thanks to my friend, Mikael HOSATTE for all the help and support I received from him. I would like to thank all my friends from the laboratory: Marc BEURET and Cedric PERRETON for their constant help and our great discussions. Lastly, I would like to thank the invited members of the committee, Prof. Marie-Catherine PALAU and M. Federico SPRENG from Astrium Space Transportation, for their interest and encouragement.. ..

(9) General introduction.. 1. Chapter I. Theory of photovoltaic conversion. 6. 1. The Sun. Solar Energy.. 7. 1.1.. Solar spectrum.. 7. 1.2.. Energy fluxes and concentration.. 9. 1.3.. Air Mass.. 12. 2. Silicon as a semiconductor.. 13. 2.1.. Properties of crystalline silicon.. 15. 2.2.. Impurities and defects in Si.. 27. 3. Theoretical limits of photovoltaic conversion.. 35. 3.1.. Thermodynamical limit of solar energy.. 35. 3.2.. Efficiency limit in a quantum converter.. 41. 4. Semiconductor solar cells.. 46. 4.1.. Properties of a p-n junction.. 46. 4.2.. Principles of operation.. 51. 4.3.. Theoretical efficiency limit in Si solar cell.. 55. 4.4.. Next generation of light to electricity converters.. 62. Chapter II. Concept of MIND structures.. 74. 1. Basic principles and device architecture.. 77. 1.1.. Theoretical conversion limit.. 78. 1.2.. Conversion limit for realistic device architecture.. 86. 2. Realization of test devices.. 91. 2.1.. Fabrication stages.. 92. 2.2.. Process key points.. 95. 3. Conclusions.. I|Page. 96.

(10) Chapter III. Numerical simulations.. 98. 1. Ab initio simulations of some properties of silicon.. 101. 2. Optical functions of crystalline and amorphous Si.. 108. 2.1.. Harmonic oscillator approximation.. 111. 2.2.. Drude-Lorentz model in a limited energy range.. 115. 3. Transition Matrix Approximation. Simulation of a 1D multi-layered structure.. 123. 3.1.. Interface between amorphous/crystalline Si. Effective Medium Theory.. 132. 3.2.. Optical simulations of MIND structures.. 136. 3.3.. Influence of spot position on reflectivity. Role of the electrodes.. 140. 4. Collection Efficiency and energy distribution. Poynting vector.. 144. Chapter IV. Experimental characterization of MIND test structures.. 154. 1. Structural measurements.. 157. 1.1.. Backscattered Electron Microscopy.. 157. 1.2.. Surface morphology.. 169. 2. Optical measurements.. 173. 2.1.. Reflectivity in the visible and near infrared.. 173. 2.2.. Ultrafast spectroscopy of chosen MIND structures.. 186. 3. Quantum Efficiency measurements. 195. 4. Electrical measurements.. 207. 4.1.. Analysis of the existing structures.. 208. 4.2.. Optimized MIND structures.. 210. II | P a g e.

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(12) The history of mankind is also a history of human energy usage. Both progressed from human muscle power and firewood in the prehistoric times, and later, thanks to agricultural surpluses, also from animal power. Humanity was slowly crawling out from the stage of barbarism, harnessing the power of wind and water, which allowed our history to advance more rapidly, up to the pre-industrial era. At some point in our history we fully harnessed the power from the fossil fuels: coal, gas and oil, thanks to the invention of steam engine; those are the times of Industrial Revolution. Over that period, roughly from 1750 until 1850, world's population increased sixfold, resulting in even more dramatic energy demand. Before that period, 70% of total work was delivered by human muscles, most of the rest by domestic animals, but in the times of Industrial Revolution, fossil fuels were the only present energy source that could fulfill the constantly increasing demand. Coal fueled the industrial revolution in the 18th and 19th century. With the advent of the automobile, airplanes and the spreading use of electricity, oil became the dominant fuel during the twentieth century. The growth of oil as the largest fossil fuel was further enabled by steadily dropping prices from 1920 until 1973.. Figure 1. Worldwide possible coal production. Predicted production peak will occur in late 2020s. Today, in the post-industrial era, our consumption of fossil fuels and electricity is enormous. On one hand fossil fuels are abundant, effective and easy to transport and those features 1|Page.

(13) allowed us to reach such an incredible level, but on the other hand humanity used more energy in the 20th century only than is used in all of the rest of human history. From that perspective an obvious conclusion makes itself apparent; fossil fuels cannot and will not last forever, therefore a new energy source must be implemented in their place, if we humans want to maintain the same life-style and civilization level. There have been several initiatives taken and several candidates are present, notably the atomic power, which is by far the most. Figure 2. World Gross Domestic Product (GDP) and energy consumption for the whole recorded history. The explosion in 19th century is linked with industrial revolution. efficient way of producing electricity, but is also the one that causes the most trouble. Even now, when we do not rely fully on it, one relatively small malfunction of the atomic reactor in a power plant, such as the one that happened in Chernobyl on 26 April 1986, or the one at Fukushima in 2011, could have catastrophic effects. The renewable energy sources offer the unique possibility of electric power production without any major drawback and renewable energy sources are abundant enough to live up to the constantly increasing world's energy demand.. 2|Page.

(14) Figure 3. World energy consumption. Only 16% is supplied from renewable sources and only 1.5% from solar energy. Today world's energy consumption is somewhere above 19 TW, while the potential of renewable sources is: 32 TW geothermal, 870 TW wind power and 86 000 TW solar power. It is easy to note that any of those energy sources is capable of fulfilling the world's need now and in the near future, with solar power being the most abundant and most accessible.. Figure 4. Potential global availability of renewable energy sources and global annual energy consumption. Sun power offers more than 5600 times energy than global needs of today. The Sun is an average star. It has been burning for more than 4-bilion years and will burn at least that long before turning into red giant, engulfing the Earth in the process. The Sun is responsible for nearly all the energy available on Earth. The exceptions are attributable to. 3|Page.

(15) moontides, radioactive material and the Earth's residual internal heat. Everything else is converted form of a Sun's energy: Hydropower is made possible by evaporation-transpiration due to solar radiant heat; the winds are caused by Sun's uneven heating of Earth's atmosphere; fossil fuels are remnants of organic life previously nourished by the Sun; and photovoltaic electricity is produced directly from sunlight by converting the energy in sunlight into free charged particles within certain kinds of materials. Figure 5. Global photovoltaic market. A rapid growth in production is observed for the past few years. In theory, with today’s knowledge, humanity has all the necessary tools in hand to rely completely on renewable energy sources while maintaining the highly developed civilization and current growth rate. It can be argued that, even if the fossil fuels triggered the events that allowed for such a dramatic changes that occurred in 19th and 20th century, regarding technological advancement, transformation of the society and rapid increase in human population, those fossil sources are not capable to upkeep this tendency and if no other energy source is found, human civilization will collapse and reverse a couple of hundreds of years in development. Therefore there are two fundamental reasons to study intensively the potential of renewable energy sources and solar power in general. The first is to maintain the current state of civilization and development while the second reason is related directly to our current status on the Kardashev scale. The Kardashev scale is a method of measuring a civilization's level of technological advancement, based on the amount of energy a civilization is able to utilize. The scale has three designated categories called Type I, II, and III. A Type I civilization uses all available resources impinging on its home planet, Type II harnesses all. 4|Page.

(16) the energy of its star, and Type III of its galaxy. At the present state, the humanity is a type 0 civilization, but according to some scientist, like Prof. Michio Kaku and Prof. Steven Hawking, we will experience the transfer to a planetary, type I civilization in the next 50 years. If this is to happen and if we are to ensure our further growth in technological advancement, there is no other long term choice than to embrace, accept and utilize the immense amount of energy the Sun is giving us every day.. 5|Page.

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(18) The Sun is the star at the center of our planetary system. It is almost perfectly spherical and consists of a hot plasma interwoven with magnetic fields. It has a diameter of about 1 392 000 km, about 109 times that of the Earth, and its mass (about 2×1030 kilograms, 330 000 times that of the Earth) accounts for about 99.86% of the total mass of the Solar System. Chemically, about three quarters of the Sun's mass consists of hydrogen, while the rest is mostly helium. The remaining (1.69%, which nonetheless equals 5 628 times the mass of Earth) consists of heavier elements, including oxygen, carbon, neon and iron, among others. The mean distance of the Earth from the Sun in about 149.6 million kilometers, which is equal to 1 AU. At the average distance, light travels from the Sun to the Earth in about 8 minutes 19 seconds. In the stellar classification, the Sun is a G2V class star, meaning that the major part of its radiation is in the yellow-green portion of the spectrum, has surface temperature about 5778 K and is a main-sequence star, generating its energy through nuclear fusion of hydrogen into helium. The total power emitted by sun equals 3.846×1026 W, which corresponds to a mean intensity of 2.009×107 W·m−2·sr−1 [1]. The Sun releases 95% of its output as light, while the remaining 5% consists of highly energetic X-rays and radio signals. Since the peak of radiation is in the green portion of the visible spectrum, most plants and the human eye function best in green light through adaptation to the nature of the sunlight reaching them.. The spectrum of the Sun's solar radiation is close to that of a black body with a temperature of about 5778 K. The Sun emits EM radiation across most of the electromagnetic spectrum. Although the Sun produces Gamma rays as a result of the nuclear fusion process, these very high energy photons are converted to lower energy photons before they reach the Sun's surface. Therefore the Sun emits electromagnetic radiation in the X-ray, UV, visible, infrared range and also radio waves. Most of the high energy photons are absorbed by the Earth's atmosphere before they reach the Earth’s surface, such as the most UV light. The spectrum of electromagnetic radiation striking the Earth's atmosphere spans a range of 100 nm to about 1 mm. It can be divided into five regions in increasing order of wavelengths:. I.7 | P a g e.

(19) - Ultraviolet or UV, divided into three parts: C: 100 to 280 nm, B: 280 to 315 nm and A: 315 to 400 nm. UVC and UVB radiation is greatly absorbed by the atmosphere and is responsible for the photochemical reaction leading to the production of the ozone layer. The least dangerous part of the UV light is the UVA, which is often used for tanning and treatment of various skin diseases. - Visible range commonly referred to as light spans 380 to 780 nm. It is the only part of the spectrum that is visible to the naked human eye. - Infrared or IR, divided also into three parts: A: 700 nm to 1400 nm, B: 1400 nm to 3000 nm and C: 3000 nm to 106 nm. It is responsible for an important part of the electromagnetic radiation reaching the Earth.. Figure I.1. Solar spectrum at the top of the atmosphere, at the sea level and the radiation of a black body with temperature T = 5778 K [2]. The Total Sun Irradiation (TSI) upon the Earth, measured by the Solar Radiation and Climate Experiment (SORCE) satellite is estimated to be around 1361 W/m² [3].. I.8 | P a g e.

(20) To understand the origin of sunlight one can consider the Sun as a black body with a temperature Ts = 5778 K [4], since this is the temperature of the Sun's surface. The number of photons emitted by a unit of volume of such a black body can be expressed by the Planck's law: (I.1.1) where u(ω,T) is the spectral density,. is the reduced Planck constant, c is the speed of light. and k is the Boltzmann constant. Energy of photons emitted by a unit of volume is described by: (I.1.2) So the total energy emitted by a given volume is expressed by: (I.1.3). !. (I.1.4). The total number of photons emitted is then given by: "# $. (I.1.5). and their mean energy is equal to:. %. &. "#'$. (I.1.6). For the Sun, at the temperature Ts = 5778 K we obtain:. %. &()*. #+. ", -. (I.1.7). If ephΩ is the energy density for a unit of a solid angle, then the energy flux emitted by a surface element dA of a solid angle comprising a source would be:. I.9 | P a g e.

(21) ./. 0 0 10. (I.1.8). Surface elements of the energy emitter (Sun) dAS and the energy receiver dAE are presented on figure I.2.. Figure I.2. Surface elements dAS and dAE of the source (Sun) and receiver (Earth) of energy, respectively. If the distance between the Sun and the Earth is RSE, the solid angle is defined by: 1( / 3(/. 2( /. (I.1.9). Assuming all the energy emitted by the Sun reaches the Earth, the energy density per solid angle emitted by the Sun is equal the energy density per solid angle received by the Earth: 564. (4. 0. /4 4. (I.1.10) (I.1.11). and energy density is equal to:. 56. 564 0. (I.1.12). If we approximate the Sun as a perfect sphere, the total energy flux emitted by the Sun in the solid angle dΩ is equal:. I.10 | P a g e.

(22) .6(. 564 0. 0. 0 78. (I.1.13). where RS is the radius of the Sun. Following the Stefan-Boltzmann law, the total power. emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:. 59. :. (I.1.14). so the total power emitted by the Sun in given by:. ;(. 78:. (. (I.1.15). Since the Sun emits that power evenly in all directions, the fraction of power that strikes the Earth is then given by the following expression:. ;(/. ;(. 7< 7 8<. (I.1.16). where RE is the radius of the Earth. Every real planet reflects part of the incident radiation. The amount of power reflected by the planet is described by its albedo α. In other words, Earth absorbs a fraction 1 - α of the Sun's light and reflects the rest. The power absorbed is given by:. ;=>?. @ ;(/. (I.1.17). even if the planet absorbs only a circular area, it emits equally as a sphere. When a planet, considered as a black body, is thermal equilibrium with its surroundings, it emits exactly the same amount of energy it receives from the Sun. The planet emits mainly in the IR part of the emits AB of the radiation that a black body would emit, where AB is the average emissivity in the spectrum, since its temperature is much lower than that of the Sun. In this frequency range it. IR range. The power emitted by the planet is then given: ;6CD. AB. 7E:. F. (I.1.18). Substituting the expressions for solar and planet power in equations I.1.14-I.1.18 and simplifying yields, the estimated temperature of the planet, ignoring the greenhouse effect is given by:. I.11 | P a g e.

(23) F. (. H G3(. AB. "3(/. @. (I.1.19). Assuming α = 0.306 [5], RS = 6.98 108 m, RSE = 1.496 1011 m [6], simplifying AB to 1 and ignoring the greenhouse effect, the average temperature of the Earth is: "! #+!IJ. F. KLM. (I.1.20). The difference between the real and estimated temperature is mainly due to the greenhouse effect. For several applications, solar power concentrated with the help of lenses and mirrors might be used. Fulfilling the assumptions of the energy balance, we have: 7 8 5(. 7 < 5/. (I.1.21). which comes from the second law of the thermodynamics, stating that the radiative temperature of the concentrated light cannot exceed the radiative temperature of the Sun. The above expression with the concentration becomes:. N5/. :. O. P:. (. 5(. (I.1.22). Therefore the maximum concentration becomes:. NC=Q. 5( 5/. R. 3( S 3/. I I. (I.1.23). The amount of light that reaches the ground is influenced by both the elliptical orbit and the Earth's atmosphere. The extraterrestrial solar illuminance Eext, corrected for the elliptical orbit by using the day number of the year is given by the following expression [7]: T6QD. T?U V W X 0 YZ R". and. I.12 | P a g e. [\ + S] +I!. (I.1.24).

(24) X. ^. 3 3=. $#$++. ". (I.1.25). being the ratio between Earth's perihelion and aphelion squared. The day-3 term in equation (I.1.24) comes from the fact that in modern times Earth's perihelion occurs around January 3rd each year. The solar illuminance constant Esc = 128 103 lx, the direct normal illuminance Edn, corrected for the attenuating effects of the atmosphere is given by:. T_*. T6QD. `UC. (I.1.26). where c is the atmospheric extinction coefficient and m is the relative optical air mass. The term air mass normally indicates relative air mass, the path length relative to that at the zenith at the sea level, so by definition, at the sea level air mass is equal to 1. With increasing angle between the source and the zenith, the air mass increases also, reaching the value of approximately 38 at the horizon. Air mass can be less than one at the elevation greater than the sea level, but most of the closed form expressions do not include that effect. In some disciplines the air mass is indicated by an acronym AM; additionally the value is given by appending the value to the acronym, AM1 indicates an air mass of 1 and so on. The region above the Earth's atmosphere, where there is no atmospheric attenuation is considered to have air mass 0 (AM0).. Silicon (latin: silicium) is the chemical element that has the symbol Si and atomic number 14 (column 4). A tetravalent metalloid, silicon is less reactive than its chemical analog carbon. Silicon is the eighth most abundant element in universe by mass. It occasionally occurs in nature as the free element, but is more widely distributed in dusts, planetoids and planets as various forms of silicon dioxide and silicates. Silicon is the second most common element of the Earth’s crust, making up 25.7% of its mass. Silicon has found many applications in industry, especially in electronics. The use of silicon in semiconductor devices demands a much greater purity than afforded by metallurgical grade silicon. Very pure silicon (>99.9%) can be extracted directly from solid silica or other silicon compounds by molten salt electrolysis. Though this method was known as early as in 1854, the rapid expansion of silicon technology and especially silicon solar technology has been I.13 | P a g e.

(25) made available thanks to Czochralski method in 1916. To this day this method is the one most widely used in industrial scale silicon production and is the cheapest one available. Elemental silicon is the principal component of most semiconductor devices such as photodiodes, integrated circuits or microchips. Silicon is also the most frequently used semiconductor, because unlike many other semiconductors it maintains its properties over a wide temperature range. Another reason for using this semiconductor is that its native grown dioxide can be easily obtained in a furnace and creates a very good semiconductor-dielectric interface and electronic passivation surface. In the form of silica (silicon dioxide) it forms various glasses, ceramics and cements, which are used in many branches of the industry. It can also be used for the creation of synthetic glass-like compounds, containing silicon, carbon, oxygen and hydrogen, named silicones. Silicon is also an essential element in the living world. While it is mainly required by plants, only small traces of it appear to be needed by animals. Silicon is an elemental semiconductor. There is no simple definition of semiconductors, but one can apply the name to materials whose electrical conductivity σ lies between 10-9 to 102 Ω-1cm-1. Different from metals, where the conductivity decreases with temperature, the conductivity of semiconductors for some temperatures increases exponentially. This is caused by the exponential increase of carrier concentration with temperature (thermal generation), as predicted by equations (I.2.22) and (I.2.23). Apart from that, the conductivity of semiconductors depends strongly on additional effects like doping or illumination. Considering band structure, it is assumed that the band gap for semiconductors lies between 0 and 3 eV. Materials for which the band gap is greater than 3 eV (while conductivity is lower than 10-9 Ω-1cm-1) are called insulators. This is a limited definition, because for different reasons one can call diamond-structured carbon a semiconductor, which has band gap of 6eV, or semi-insulator GaAs. The reason these materials are considered semiconductors is due to the type of conductivity. Metals have only electrons, semiconductors have electrons and holes, but lack ionic conductivity, visible in dielectrics. In semiconductors the most important role is always played by electron-hole conductivity, before the ionic one. On the other hand, semiconductors with 0 eV effective bang gap are called semimetals.. I.14 | P a g e.

(26) ! At T = 298 K (room temperature) silicon exists in the solid state. It can exist in other phases such as crystalline, amorphous and polycrystalline. The melting point for silicon is at T = 1687 K and its boiling point is at T = 3173 K.. Figure I.3. Crystalline silicon forms in a diamond structure, space group Fd-3m. The cell parameters for Si are 543.09 pm (picometre). The electronic configuration of silicon is [Ne].3s2.3p2 where [Ne] stands for structure of neon [8]. To understand the origin of properties of crystalline silicon one must consider a silicon crystal extended to infinity in all three dimensions. Properties of such a crystal can then be analyzed by the Bloch's theorem [9], which states that the eigenstates ψ of the one-electron Hamiltonian: a. b Wd e "c. (I.2.1). where U(r+R) = U(r) for all R in a Bravais lattice, can be chosen to have the form of the plane waves times a function with the periodicity of the Bravais lattice: f*g e. hg0i. *g. e. (I.2.2). where unk(r+R) = unk(r) for all R in the Bravais lattice. the equations (I.2.1) and (I.2.2) imply. that:. I.15 | P a g e.

(27) f*g e W 3. hg0j. f*g 3. (I.2.3). By imposing appropriate boundary conditions, such as Born-von Karman conditions [9], we can demonstrate that the k vector must be real and arrive at a condition restricting the values of k. If we are working with the case where the primitive cell is no longer cubic, we can generalize the boundary conditions as follows:. f e W kh [h. f e l. "+. (I.2.4). where the ai are three primitive vectors and the Ni are all integers of order N1/3, where N = N1N2N3 is the total number of primitive cells in the crystal. Applying Bloch's theorem. (I.2.3) to the boundary condition (I.2.4) we find that: f*g e W kh [h. hmn g0=n. which requires that: l. hmn g0=n. W. (I.2.5). "+. When k has the form: o po. f*g e. p W. (I.2.6) p. where pq 0 [q. " rhq. (I.2.7). and aj, bj are the normal and reciprocal lattice vectors, respectively, equation (I.2.6) requires the following: hmn Qn. (I.2.8). and consequently we must have: h. ch kh. (I.2.9). mi integral. Therefore the general form for allowed Bloch wave vectors is: s hto. ch p kh h. (I.2.10). The volume of a reciprocal lattice primitive cell is (2π3)/v, where v = V/N is the volume of a direct lattice primitive cell, so the volume ∆k of k-space allowed value of k can be expressed as follows:. I.16 | P a g e.

(28) u. " -. (I.2.11). The ground state on N Bloch electrons is constructed by occupying all Bloch levels labeled by the quantum numbers n and k, but, unlike the free electron case, εn(k) does not have a simple explicit free-electron form and k must be confined to a single primitive cell of a reciprocal lattice if each level is to be counted only once. When the lowest of these levels are filled by a specified number of electrons, two quite distinct types of configuration can result: - A certain number of bands may be completely filled, all others remain empty. The difference in energy between the highest occupied level and the lowest unoccupied level is known as the band gap. We shall find that solids with a band gap greatly in excess of kBT (T near the room temperature) are insulators. If the band gap is comparable to kBT, the solid is known as an intrinsic semiconductor. Because the number of levels in a band is equal to the number of primitive cells in the crystal and because each level can accommodate two electrons (one of each spin), a configuration with a band gap can arise only if the number of electrons per primitive cell is even. - A number of bands may be partially filled. When this occurs, the energy of the highest occupied level, the Fermi energy εF, lies within the energy range of one or more bands. For each partially filled band there will be a surface in k-space that separates occupied from the unoccupied levels. The set of all such surfaces is known as the Fermi surface and is the generalization to Bloch electrons of the free electron Fermi sphere. The parts of the Fermi surface arising from individual partially filled bands are known as branches of the Fermi surface. A solid has metallic properties provided that a Fermi surface exists. One must very often calculate quantities that are weighted sums over the electronic levels of various on-electron properties. Such quantities are of the form:. v. " s v* *g. (I.2.12). where for each n the sum is over all allowed k giving physically distinct levels. In the limit of a large crystal the allowed values of k get very close together, and the sum may be replaced by an integral. Since the volume of k-space per allowed k has the same value as in the free electron case, the prescription for the free-electron model case remains valid and we find that:. I.17 | P a g e.

(29) w. xyz. {|. v -. "s. ". *. v*. (I.2.13). where the integral is over the primitive cell. If v*. depends on n and k only through the. energy εn(k), then in further analogy to the free electron case one can define a density of states per unit volume g(ε) so that q has the form: w. }~ } v }. (I.2.14). Comparing equations (I.2.13) and (I.2.14) we find that:. k }. s k* } *. (I.2.15). where Nn(ε), the density of states in the nth band is given by: k* }. r }. }*. (I.2.16). where the integral is over any primitive cell. In an intrinsic semiconductor, the number of occupied conduction band levels is given by [10]:. €•‚ƒ. €„. k } • }. }. (I.2.17). where εC is the energy at the bottom of the conduction band and εtop is the energy at the top. The density of states N(ε) can be approximated by the density near the bottom of the conduction band for low-enough carrier densities and temperatures (the so called parabolic band approximation): k }. …U. †" ‡. ‡O. o. c_6. ˆ. (I.2.18). where MC is the number of equivalent minima in the conduction band and mde is the density of state effective mass for electrons:. c_6. co9 c9 c9. oˆ. (I.2.19). where co9 , c9 and c9 ,are the effective masses along the principal axes of the ellipsoidal. energy surface. The Fermi-Dirac distribution function F(ε) is given by:. I.18 | P a g e.

(30) • }. ‡. W. Š. ‡‰. (I.2.20). where kB is the Boltzmann constant, T absolute temperature and εF the Fermi energy. The integral in the equation (I.2.17) can be evaluated to be: kO. ". †. •oˆ R. ‡‰. •O. Š. S. (I.2.21). where NC is the effective density of states in the conduction band and is given by: kO. " c_6 "R ‹. Š. ˆ. S. …O. (I.2.22). and F1/2(ηf) is the Fermi-Dirac integral. For the Boltzmann statistics case, that is for the Fermi ˆ" and equation (I.2.21) becomes:. level several kBT below EC in non-degenerate semiconductors, the integral approaches. †. Œ•. kO. ‡O. R. Š. ‡‰. S. (I.2.23). Similarly, we can obtain the hole density near the top of the valence band: k{. ". †. •oˆ R. ‡{. ‡‰. Š. S. (I.2.24). where Nv is the effective density of states of the valence band and is given by: k{. " c_ "R ‹. Š. ˆ. S. (I.2.25). where mdh is the density of state effective mass of the valence band: c_. cŽ W c 9. 9. ˆ. (I.2.26). where subscripts refer to light and heavy holes, respectively. Under non-degenerate conditions we obtain: k{. R. ‡{. Š. ‡‰. S. (I.2.27). As can be seen from the above equations, knowledge about the density of states is vital for calculating the real number of free-carriers in given conditions.. I.19 | P a g e.

(31) Equations (I.2.22) and (I.2.25) provide good approximation for the points in the band maxima and minima and their surroundings, but often more detailed description is needed. Such description can be provided by the Density Functional Theory methods or the Local Density Approximation [11,12]. Having determined the Density of States one can try to estimate the position of Fermi level in intrinsic semiconductors. In such materials, at finite temperatures continuous thermal agitation exists, which results in excitations of electrons from the valence band to the conduction bands and leaves an equal number of holes in the valence band, that is, n = p = ni, where ni is the intrinsic carrier density. This process is balanced by recombination of the electrons in the conduction band with holes in the valence band.. Figure I.4. The Electronic Density of States of crystalline silicon calculated by ab initio method within the frames of Local Density Approximation. Characteristic van Hove singularities can be seen. The Fermi level for an intrinsic semiconductor is then obtained by equating (I.2.23) and (I.2.27):. I.20 | P a g e.

(32) }‰. }h. ‡O. ". ‡{. W. k{ • R S " kO. Š. ‡O. ". ‡{. W. +. Š. c_ • • ‘ ˆ c_6 …U. (I.2.28). Hence the Fermi level for an intrinsic semiconductor (for T close to 293 K) generally lies very close to the middle of the band gap.. Figure I.5. Intrinsic carrier density in silicon vs T, calculated using the density of states presented in figure I.4. One can gain a substantial insight into the structure imposed on the electronic levels by a periodic potential, if that potential is rather weak. There are two main reasons behind that: the electron-ion interaction is the strongest at small separations, but the conduction electrons are forbidden from entering the immediate neighborhood of the ions by the Pauli principle, since this region is already occupied. Second reason is that in the area where the conduction electrons are allowed, their mobility further diminishes the net potential any single electron experiences, for they can screen the fields of positively charged ions. When the periodic potential U = 0, the solution to the Schrodinger's equation are plane waves. The wave function of a Bloch level with crystal momentum k can be written as follows:. I.21 | P a g e.

(33) fg e. s g. h g`’ 0i. g`’. (I.2.29). where the coefficients ck-K and the energy levels ε are determined by the set of equations: “ "c. J. }”. g`’. W s d’•`’ ’•. g`’•. $. (I.2.30). The sum in (I.2.29) is over all reciprocal lattice vectors K, and for any fixed k there is an equation of the form (I.2.30) for each reciprocal lattice vector K. The infinitely many solutions of equation (I.2.30) are labeled with band index n. When periodic potential U is zero and we are dealing with free electron case, the solution divides naturally into two possibilities: the non-degenerate case, where there is only one K vector for which the equation (I.2.30) is satisfied: }. }g`’ fg –. h g`’ i. (I.2.31). and the degenerate case, if there is a group of reciprocal lattice vectors K1,...,Km satisfying:. }g`’—. ˜. }g`’™. (I.2.32). Situation becomes more complex when U is no longer 0, but very small. There are two. possible scenarios: Scenario 1. We fix k and consider such reciprocal lattice vector K1 that: š}g`’—. }g`’ š › d,. for fixed k and all K K1. (I.2.33). and we wish to investigate the effect of U on free-electron level given by:. }. }g`’—. g`’. $,. K. K1. (I.2.34). Setting K = K1 we have:. œ}. }g`’— •. g`’—. s dg`’— ’. g`’. (I.2.35). Because we chose the additive constant in the potential energy so that UK = 0 when K = 0, only terms with K. K1 appear on the right-hand side of equation (I.2.35). We consider a case. where ck-K vanishes when K K. K1 we obtain:. I.22 | P a g e. K1 in the limit of vanishing U, so writing equation (I.2.30) for.

(34) d’— `’ }. g`’. g`’—. }g`’. W s. ’ž’— ,. d’•`’ g`’• } }g`’. (I.2.36). Therefore, provided that there is no near degeneracy (which could cause some of the denominators in (I.2.36) to be of order of U and resulting in additional terms in the sum to be comparable to the term K = K1) we can write: d’— `’ }. g`’. g`’—. }g`’. WŸ d. (I.2.37). Combining this with (I.2.35) we find: œ}. }g`’— •. s. g`’—. ’,. d’— `’ d’`’— }. }g`’. g`’—. WŸ d. (I.2.38). Thus, as expected the perturbed energy level ε differs from the free-electron value }g`’— only. by the order of U2. To solve equation (I.2.38) for ε, it suffices to replace the appearing ε in the denominator on the right-hand side by }g`’— leading to the following expression, correct to. second order in U: }. }g`’— W s ’,. šd’`’— š. }g`’—. }g`’. WŸ d. (I.2.39). every energy level }g`’ that lies below }g`’— contributes a term that raises the value of ε,. Equation (I.2.39) asserts that weakly perturbed, non-degenerate bands repel each other, for while every energy level that lies above }g`’— contributes a term that lowers the value of ε.. Also, in the case of no near degeneracy, the shift in energy from the case of the free-electron value is second order of U. K1,..., Km and corresponding energy states }g`’— ,..., }g`’™ all within order U of each other,. Scenario 2. We suppose that the value of k is such that there are reciprocal lattice vectors but far apart from other energy states }g`’ on the scale of U: š}g`’. }g`’n š › d,. i = 1, ... , m,. K K1, ... , Km. (I.2.40). In this scenario we must treat separately all m equations of type (I.2.30) for any given K equal to K = K1,..., Km. In these m equations we separate from the sum all the terms containing. I.23 | P a g e.

(35) coefficients ck-Kj, j = 1... m, which need to be small in the limit of vanishing interaction, from the remaining ck-K that will be at most of the order of U. We obtain: œ}. }g`’n •. C. s d’. g`’n. qto. `’n g`’. W. s. ’ž’— ¡ ’™. dg`’n. g`’. (I.2.41). for i = 1, ..., m. If we assume the same separation in the sum, we obtain: }. g`’. for K. }g`’. C. ¢s d’ qto. `’ g`’. W. s. ’ž’— ¡ ’™. d’£ `’. ’`’• ¤. (I.2.42). K1,..., Km, which corresponds to the equation (I.2.36) in the non-degenerated case.. Knowing that ck-K will be at most of the order of U for any K becomes: g`’. }. }g`’. C. s d’ qto. `’ g`’. K1,..., Km, equation (I.2.42). WŸ d. (I.2.43). substituting equation (I.2.43) into (I.2.41) we find that:. œ}. }g`’n •. g`’n. C. s d’ qto. `’ g`’. C. W s¢ qto. s. ’ž’— ¡ ’™ ,. d’— `’ d’`’— }. }g`’. ¤. g`’. WŸ d. (I.2.44). We found that the shift in m nearly degenerate levels reduces to the solution of m coupled equations for ck-Kj. Since coefficients in the second term on the right-hand side are of the higher order than those in the first, one can find the leading corrections in U are: œ}. }g`’n •. g`’n. C. s d’ qto. `’ g`’. (I.2.45). for i = 1, ..., m, which are just set of equations for a system of m quantum levels. The quantum structure of real solids is usually so complex that the nearly free-electron model is rarely valid [9]. One obvious observation that arises is the complete disregard of bands arising from ion-core levels. Several methods exist that allow rigorous analysis of core levels and resulting bands, the most-widely used one being the Tight-Band method. Nevertheless, cases where the interest is directed towards either deep core or highly excited, near freeI.24 | P a g e.

(36) electron states are rather rare. Therefore several combined methods have been introduced that allow accurate representation of band structure for both areas of interest. Methods that gained most attention are: Independent Electron Approximation, Cellular Method, Muffin-Tin Potential, Augmented Plane Wave Method, Green's Function Method, Orthogonalized Plane Wave Method and Pseudopotential Method [13]. It is not the aim of this work to review all these methods, therefore we will focus briefly only on the Pseudopotential [14,15] method. The interested reader is referred to the information present in the literature [9,13]. Let us now describe the exact wave function for a valence band as a linear combination of Orthogonalized Plane Waves: fg. s. g ¥g¦’. ’. (I.2.46). and let φkv be the plane-wave part of this expansion:. ¥g§ e. s ’. ’. h g¦’ i. (I.2.47). we also keep in mind that for core levels:. ¥g. hg i. where pU. fg§ e. W s pU fgU e U. ¨ efgU9 e. ¥g§ e. sR U. (I.2.48) hg i. , then we can rewrite the expansions (I.2.48) and (I.2.46) as:. e©fgU9 e© fg§ e© S fgU e. (I.2.49). Since ψkv is an exact valence wave function, it satisfies Schrodinger's equation with eigenvalue εkv:. afg§. }g§ fg§. a¥g§. sR. (I.2.50). Substitution of (I.2.50) into (I.2.49) gives: U. e©fgU9 ¥g§ S afgU. If we note that afgU a W - j ¥g§. }g§ ¥g§. }g§ V¥g§. sR U. e©fgU9 ¥g§ S fgU ]. (I.2.51). }gU fgU for the exact core levels, then we can rewrite (I.2.51) as:. (I.2.52). where most of the cumbersome terms are confined within the operator VR, defined by: I.25 | P a g e.

(37) -jf. s }g§ U. }U R. e©fgU9 fS fgU. (I.2.53). The pseudopotential is defined to be the sum of the actual periodic potential U and VR: a W -j. "c. b W-. ?6)_ª. (I.2.54). We assume that the pseudopotential is sufficiently small to justify the nearly free-electron calculation of the valence levels. One can see a hint that this might be so from the fact that f df. actual. ¨ ef 9 e d e f e. periodic. potential. is. attractive. near. the. ion. cores. and. thus. is negative, the corresponding matrix element of the. potential VR is, according to (I.2.53): f -jf. s }g§ O. }gU «. efgU9 f«. (I.2.55). Since valence energies are above core energies, this is always positive. Thus adding VR to U provides at least partial cancellation, and one might hope for it to lead to a potential weak enough to do nearly free electron calculations for φkv, treating the pseudopotential as weak perturbation. In three dimensions the structure of energy bands is often presented by plotting ε vs. k along straight lines connecting particular high-symmetry points in the Brillouin zone. Such curves are generally shown in a reduced zone-scheme, because for general directions in k-space they are not periodic. An example of accurately calculated band structure for crystalline silicon is shown in figure I.6. The calculation was performed within the Density Functional Theory (DFT) [16,17], the Local Density Approximation (LDA) [11] [18] and Pseudopotential models. The electronic band gap for bulk crystalline Si (c-Si) is Eg = 1.12 eV. The optical band gap for bulk silicon is Eg = 1.17 eV [13].. I.26 | P a g e.

(38) Figure I.6. Electronic band structure of crystalline silicon calculated by ab initio method within LDA and DFT frames. High symmetry points are indicated on the first Brillouin zone, while the Γ point (k = 0) lies in the middle.. By a crystalline defect one generally means any region where the microscopic arrangement of ions differs drastically from that of a perfect crystal. Defects are called point, line or surface, depending on whether the imperfect region is bounded on the atomic scale to one, two or three dimensions. Particular kinds of defects that found a broad application in semiconductors are dopants (doping impurities). When a dopant is introduced in a semiconductor, electronic density and potential often introduced by that defect is often quite different than that of the surrounding ions (figure I.7). The periodic potential of a crystal is locally disturbed, resulting in additional states that appear somewhere in the energy spectrum. Depending on the nature of the defects, some of those extra states can be beneficial. Another effect that appears is that the neighboring atoms usually are no longer in their local energy minima, and the crystalline network relaxes around the defect (figure I.8). The force resulting from the introduction of a phosphorous atom and subsequent relaxation imposes a strain on the neighboring atoms. As. I.27 | P a g e.

(39) within the frames of DFT-LDA approximation by ab initio method, this pressure is equal p = 1.34904 GPa for a cell consisting of a total of eight atoms [19].. Figure I.7. Changes in the potential ∆V around a phosphorous atom substituted in Si, in regard to the potential of undoped silicon. The potential is plotted in a plane passing through the P atom and after the relaxation of crystalline network. When a semiconductor is doped with donor and/or acceptors, impurity energy levels are introduced. A donor level is defined as being neutral when occupied, and positive when empty. Likewise, an acceptor level is defined as negative when occupied and neutral when empty. The simplest calculation of impurity energy levels is based on the hydrogen-atom model. The ionization energy for the hydrogen atom is: }¬. c w +" }. +#I -. where ε0 is the free-space permittivity.. I.28 | P a g e. (I.2.56).

(40) Figure I.8. Changes in the potential ∆V around a phosphorous atom occupying an interstitial position in Si, after relaxation of the crystalline network. The potential is plotted in a plane passing through the P atom. The ionization energy for the donor εd can be obtained by replacing mo by the conductivity effective mass of electrons: cUU. + • 9 W 9 W 9‘ co c c. `o. (I.2.57). and by replacing by the permittivity of the semiconductor εs in (I.2.56): ‡_. } cU? R S R S }¬ }? c. (I.2.58). The ionization energy for donors calculated from (I.2.58) is 0.006 eV for Ge, 0.025 eV for Si, and 0.007 eV for GaAs [10]. The hydrogen-atom calculation for the ionization level for the acceptors is similar to that for the donors. We consider the unfilled valence band as a filled band plus an imaginary hole in the central force field of a negatively charged acceptor. The calculated acceptor ionization energy (measured from the valence-band edge) is 0.015 eV for I.29 | P a g e.

(41) Ge, 0.05 eV for Si, and about 0.05 eV for GaAs. The simple hydrogen-atom model given above certainly cannot account for the details of ionization energy, particularly the deep levels in semiconductors. However, the calculated values do predict the correct order of magnitude of the true ionization energies for shallow impurities. It is possible for a single atom to have many levels; for example gold in Ge has three acceptor levels and one donor level in the forbidden energy gap. The Fermi level for the intrinsic semiconductor (I.2.28) lies very close to the middle of the band gap. Figure I.9 depicts this situation, showing schematically from left to right the simplified band diagram, the density of states N(ε), the Fermi-Dirac distribution function F(ε), and the carrier concentrations. The shaded area in the conduction band and the valence band are the same; indicating that e = p = n; for the intrinsic case. When impurity atoms are introduced, the Fermi level must adjust itself to preserve charge neutrality (figure I.9 b and c). Consider the case shown in figure I.9 b, where donor impurities with a concentration ND (cm-3) are added to the crystal. To preserve electrical neutrality the total negative charges (electrons and ionized acceptors) must equal the total positive charges (holes and ionized donors), or for the present case: k-¦ W. (I.2.59). where n is the electron density in the conduction band, p is the hole density in the valence band, and ND+ is the number of ionized donors, given by: k-¦. k- ®. I.30 | P a g e. W¯. ‡-. Š. ‡‰. °. (I.2.60).

(42) Figure I.9. Schematic band diagrams, density of states, the Fermi-Dirac distribution, and the carrier concentration for: a) intrinsic, b) n-type and c) p type semiconductor at thermal equilibrium.. I.31 | P a g e.

(43) where g is the ground-state degeneracy of the donor impurity level and equals 2 because of the fact that a donor level can accept one electron with either spin or can have no electron. When acceptor impurities of concentration NA are added to a semiconductor crystal, a similar expression can be written for the charge neutrality condition and the expression for ionized acceptors is: k±`. k±. W¯. ‡±. Š. ‡‰. (I.2.61). where the ground-state degeneracy factor g is 4 for acceptor levels. The value is 4 because in Ge, Si. and GaAs each acceptor impurity level can accept one hole of either spin and the impurity level, is doubly degenerate as a result of the two degenerate valence bands at k = 0. Rewriting the neutrality condition (I.2.59), we obtain: €•‚ƒ. €„. k } • }. }. k- ®. ‡-. W¯. Š. ‡‰. °W. €². €³‚••‚™. k }. • }. }. (I.2.62a). or simply: kO. R. ‡O. Š. ‡‰. S. k- ®. W. ¯. ‡-. Š. ‡‰. ° W k{. ‡{ R. Š. ‡‰. S. (I.2.62b). For a set of given NC, NV, ND, EC, EV, ED, and T, the Fermi level EF can be uniquely determined from (I.2.62). In the case shown in figure I.9 b (with ND = 1016 cm-3, T = 300 K) the Fermi level is close to the conduction band edge and adjusts itself so that almost all donors are ionized. As the temperature is lowered sufficiently, the Fermi level rises toward the donor level (for n-type semiconductors) and the donor level is partially filled with electrons. The approximate expression for the electron density is then: R. k- k± S k"k±. R. T_ Š. S. for a partially compensated semiconductor and for:. I.32 | P a g e. (I.2.63).

(44) k± › kO ". R. T_ Š. S. where Ed = EC-ED, or: †". k- kO. oˆ. R. T_ Š. S. (I.2.64). Figure I.10 shows a typical example, where n is plotted as a function of reciprocal temperature. At high temperatures we have the intrinsic range since n = p > ND. At very low temperatures most impurities are not ionized and the slope is given by (I.2.63) or (I.2.64), depending on the compensation conditions. The electron density, however, remains essentially constant over a wide range of temperatures (T ~ 200 to 500 K in figure I.10).. Figure I.10. Carrier density vs T in P-doped silicon for a doping concentration Nd = 1018 cm-3, calculated using the density of states presented in figure I.4. When doping impurity atoms are added, the up product is still given by:. I.33 | P a g e.

(45) kO k{. R. T~ Š. S. (I.2.65). which is called the mass-action law, and the product is independent of the added impurities. At relatively elevated temperatures, most donors and acceptors are ionized, so the neutrality condition can be approximated by: W k±. W k-. (I.2.66). This analysis gives the basic insight for the charge neutrality principle and calculation of the Fermi level in semiconductors, as well as an estimation of the number of free carriers for intrinsic and doped semiconductors and their evolution with temperature.. I.34 | P a g e.

(46) The thermodynamic efficiency of the energy conversion of radiation into other forms of energy is of wide interest and has been much discussed [20,21]. The efficiency of a solar cell is defined as the ratio between incident power PS and output power of a photovoltaic device PD. The upper limit can be determined either by analyzing the thermodynamic limits of the two interacting bodies or by a detailed analysis of all the generation and recombination mechanisms, without taking into account the exact structure of the device.. Here we shall consider a system consisting of two large reservoirs called pump (P) and sink (S) together with a converter (C). The last interacts with the pump by an interchange of isotropic radiation and with the sink by isotropic radiation and possibly by other means so as to exchange work and heat. If the converter takes in black-body radiation at temperature TP from the pump and rejects black-body radiation at a temperature marginally above the sink temperature TS, then an upper limit to the conversion efficiency is [22,23]: W R + F + (. ´. (. F. S. (I.3.1). We assume that that the converter (C) of the temperature TC is encircled by two reservoirs: the Sun, which acts as a pump (P) of the temperature TP and the environment which acts as a sink (S) of the temperature TS [24]. We consider the exchange of energy and entropy C. S and C. P, and we mark je,XY and js,XY as energy and entropy fluxes from point X to Y, where X,Y = P, S, C. The schematic representation of the system considered here is shown in figure I.11.. I.35 | P a g e.

(47) Figure I.11. System Pump - Converter - Environment (PCE) with different energy and entropy fluxes and the direction of their flow. The heat conduction is not considered here, but is shown for complete description. The transfer of the heat vµ , which is done and temperature TQ (TS < TQ < TP) from the. provided by the converter and ¶µ is the generation of entropy by the unit of surface inside the. converter to the environment is associated to the transfer of entropy. W is the collectible work. converter. The balance between energy and entropy fluxes is then: 56 *6D. 56 FO. 5? *6D. 5? FO. 56 OF W 56 (O. 56 O(. 5? OF W 5? (O. 5? O(. vµ. µ ¶~6*. ·µ. vµ. (I.3.2). ¸. The efficiency of the conversion is described as follows: ¹. ·µ. 56 FO. I.36 | P a g e. œ56 FO W. ¸ 5? FO •. œ56 OF. œ56 O(. ¸ 5? OF •. ¸ 5? O( •. 56 FO. œ56 *6D 56 FO. W œ56 (O. ¸ 5? *6D •. ¸ 56 (O •. µ. ¸ ¶~6*. (I.3.3).

(48) The terms je,net and js,net are often neglected. Since the Sun is visible from the Earth under a solid angle Ω, a following assumption is justified:. 56 º» 5? º». ¼ J½º ¾¿À Áº». º». ¼ Â½º ¾¿À Áº». (I.3.4). º». with [21]: J½º Â½º. •º K. •º. ½º. Ã. K. W. ½º. xÄ. W. ½º. ½º. xÄ. ½º Å. (I.3.5). where nωX is the number of photons at frequency v = 2πω emitted by X. ΘXY is the angle that emits incident radiation normal to the surface of Y, where X, Y = S, C, P. The parameters lX = 1 for polarized light and lX = 2 for non-polarized light. If KωX and LωX are angleindependent, we can write: 56 º». Æº». J½º. 5? º». Æº». Â½º. (I.3.6). where: Æº». 4ÇÈ. ¾¿À Áº». º». (I.3.7). In the case of the PCE system, Btot = BCP+BCS represents the whole solid angle accessible to the system. We therefore obtain the following relations: Æº». ÆFO ÆDªD. Æ»º N. NC=Q. (I.3.8) (I.3.9). I.37 | P a g e.

(49) Assuming that nωX( ω) depends on parameter É temperature TX, we can obtain: !: Æ • " Ì º» º. 56 º». º. !: Æ • " Ì º» º. 5? º». º. .. Í. ½. gÊË. , and that X is a black body of. º. (I.3.10). º. where σ=5,670⋅10−8 [W⋅m-2⋅K-4] is the Stefan constant, and nX = nωX(z). Additionally:. .. º. Í. º. É. º. É. ÎxÄ. É W. º. W. xÄ R W. º. º. SÏ. É. (I.3.11). If the Pump (Sun) and the Sink (environment) are considered black bodies at temperatures TP and TS respectively, and upon assuming the same polarization factor lX for all components we obtain: º. É. Š º. (I.3.12). we can also introduce an integral G(u,v) defined as a Debye function [25]: Ð*. Ñ. ). É. É*. Ñ. É. (I.3.13). now, we find that I(nX) and J(nX) can be expressed as: .. Í. º º. Ð $$. +. Ð $$. !. !. The conversion efficiency η becomes finally:. I.38 | P a g e. (I.3.14).

(50) ¹. “. +. ¸ F. R. (. F. S W. +. NC=Q R S “R N. O. F. ¸. S. F. R. (. F. +. S ”. µ " Ì ( ¶~6* !: ÆFO •F F .. ¸ F. R. O. F. S. R. (. F. S W. F. +. ¸ F. R. (. S ”. R. (. F. (I.3.15). In the case when TQ = TS we obtain: ¹. “. +. (. F. W. +. R. (. F. NC=Q R S “R N. S ”. µ " Ì ( ¶~6* !: ÆFO •F F .. F. O. F. S. +. /. F. R. O. F. S W. +. F. S ”. (I.3.16). The maximum conversion efficiency ηL corresponds to the efficiency obtained by Landsberg and co-workers [21]. It is obtained under following conditions: − the converter is totally encircled by the Sun − the converter is in equilibrium with the surrounding environment − there is no generation of entropy inside the cell Therefore: ¹ Ò ¹´. (. +. F. W. +. R. (. F. S. (I.3.17). as in (I.3.1). It is worth noting that this efficiency is somewhat below the Carnot efficiency ¹O=i*ªD. (. F. (I.3.18). Other authors [26] proved that this difference comes from the generation of entropy when the light traverses the space between the Sun and the Earth. Another model that represents a more realistic approach to the Pump - Converter - Sink system is shown in figure I.12. This endoreversible converter takes into account irreversibility of heat transfer between different points of the system. Because of that, the maximal efficiency in that approach is always below the one obtained by Landsberg approach. The Carnot engine (CENG), coupled with a radiator (R) of the temperature TC is encircled with by. I.39 | P a g e.

(51) the Sun, which again acts as a pump (P) at the temperature TP, and by the environment, which acts as the sink (S) of the temperature TS.. Figure I.12. System Pump - Converter - Environment (PCE) and a Carnot engine representing the endoreversible converter system. In this case the conversion efficiency is expressed as: ¹. ·µ. R. 56 FO. (. F. 56 FO. S. 56 OF W 56 (O 56 FO. 56 O(. (I.3.19). by using the same notations as in previous model, we obtain: ¹. R. (. O. S“. R. NC=Q S WR S •R N F (. (. F. S. R. O. F. S ‘”. (I.3.20). For thermal equilibrium TC = TE = TP the efficiency is self-cancelling, for TC = TE and TC = TC0. TC0 corresponds to the situation where the work extracted from the converter is W = 0. Therefore we can write: O. F ^R Ó. (. F. also, we have: I.40 | P a g e. S W“. R. (. F. S ” R. N. NC=Q. S. (I.3.21).

(52) ¹. R. (. O. S. NC=Q • N. O. F. O. ‘. (I.3.22). The maximal efficiency is obtained for the temperature T, satisfying the following relation:. Ô¹ Ô O. OC. $Õ. O. Ì OC. W+. OC. (. F. (I.3.23). for the maximal concentration Cmax we obtain:. ¹. R. (. O. S •. O. F. ‘. (I.3.24). the maximal efficiency η as a function of the ratio TC/TP for different values of concentration C is shown in figure I.13.. Figure I.13. Conversion efficiency calculated from equation (I.3.20) for different concentration ratios and different converter temperature TC. TP = 5800 K and TS = 300 K.. " The maximum efficiency for C = 1 for an endoreversible converter is around 12%, which is much the below obtained efficiency for real solar cells. This is due to the fact that I.41 | P a g e.

(53) semiconductor solar cells are using only a part of the spectrum, but at elevated efficiency. To determine the efficiency in the case of a quantum converter with defined band gap, such as a semiconductor, we must take into account the quantum nature of light-matter interactions in such a system. In such case, the absorbed flux of photons is given by the Kirchoff's law of thermal radiation [27]:. [. 56 =>? a(. (I.3.25). Š. ) is the absorption degree. For a sufficiently thick cell, we assume that a = 0 for any < Eg and a = 1 for all other wavelengths. Photons with energy inferior to the band gap are. not absorbed by the converter. Photons emitted by the Sun and environment have a chemical potential equal to zero, while photons emitted by the converter have a chemical potential equal to qV. Therefore we obtain: 56 º» 5? º». with:. !: Æ • " Ì º» º. !: Æ • " Ì º» º. F. R. as well as: O. º. .. Í. º. (I.3.26). º. (I.3.27). Š F. and: (. º. R. Š (. S. Š O. S. upon assuming that. (I.3.28). (I.3.29) /Ö. gÊ×. ,Ñ. Ø{. gÊÙ. µ and recalculating equations (I.3.11) and neglecting ¶~6*. we find the maximum output efficiency:. I.42 | P a g e.

(54) ¹C=Q. œ56 FO. ·µ ;h*. ¸ 5? FO • W. œ56 (O. ¸ 56 (O •. ;h*. œ56 OF. ¸ 5? OF •. œ56 O(. ¸ 5? O( •. (I.3.30). or incorporating the function G (equation I.3.13) we get: ¹C=Q. !. ÚR. +. WR W. wS Ð. NC=Q •( S [ ÎR N •F. NC=Q •O p ÎR N •F. wÑ Ð p. w. $. Ñ ÏÛ p p. +. w SÐ +p. xÄ. w SÐ +[. Ñ p p. `). [. $. w +p p. w +[ [ xÄ. xÄ. `. §`) >. `. ) =. Ï. (I.3.31). Even more accurate description that includes the degree of polarization and reflectivity for any photon absorbed or emitted by the converter is given by Badescu and Landsberg [28]. We have shown that, quite surprisingly, efficiency of an endoreversible photovoltaic converter under normal, natural irradiation (C = 1) is only 11.7%. It has been shown that real solar cells have largely exceeded this value [29]. The difference comes from different mode of conversion employed. In the endoreversible thermodynamical converter the whole spectrum is absorbed and used, while in the photovoltaic converter absorption is selective and only photons with energy E. Eg are absorbed and converted, but with different efficiency.. In 1961, Shockley and Queisser first introduced the idea of detailed balance limit analysis for a solar cell [30], by assuming several hypotheses: − Solar cell is transparent for any photon with energy below the band gap Eg, − All photons with energy higher or equal to Eg are always absorbed and always create one electron-hole pair, − Carrier mobility is infinite, − There are two possible types of recombination: radiative and non-radiative. The parameter ρ is defined as:. Ü. e[ l[ÝlÑ ,e Ycpl [ÝlY ,e[Ý ÝYÝ[•,e Ycpl [ÝlY ,e[Ý. (I.3.32). I.43 | P a g e.

(55) In such case, the total current density J flowing through a photovoltaic cell is given by the difference between carriers generated by external illumination and recombination inside the cell. The detailed balance equation is then gives as: Í. 5* FO W 5* (O. 3O. (I.3.33). where jn,PC is the number of photons emitted by the pump (Sun) and absorbed by the cell, jn,SC is the number of photons emitted by the sink (Environment) and absorbed by the cell, RC is the total recombination rate and e is elemental charge. In our case we assume the both pump and sink can be considered as black bodies, therefore we obtain: 5* FO. N. NC=Q. 5* (O. R. ". N. NC=Q. S. Ð R. Š F. ". Š O. T~. Š F. $S. Ð R. T~. Š O. $S. (I.3.34). Contrary to the endoreversible converter case, the integration is done over T Þß T~ à á.. Total recombination rate is given by: 3O. Ü. 5* O. (I.3.35). after generalized Planck's law we obtain: 5* O. ". Š O. Ð R. T~. Š O. w-. Š O. S. (I.3.36). conversion efficiency is then expressed as follows: ¹. . âh*. (I.3.37). where Πin is expressed as: âh*. N. NC=Q. ". by substituting X. Š F. ÜO. O. ™ãË. Ð $$. , we can express the Shockley-Queisser efficiency (SQ) for the case. in which the pump and the sink are considered black bodies:. I.44 | P a g e. (I.3.38).

(56) ¹(¸. !. -. Š F. äÐ R X. R. O. F. w-. Š F. $S. S VÐ R. R. T~. Š O. O. F. S Ð R -. Š O. S. T~. Š O. Ð R. $S. T~. Š O. $S]å. (I.3.39). Figure I.14. The photovoltaic efficiency surface η (Eg,V). Pump (Sun) at TP = 5800 K, concentration factor Cmax = 46000, radiative efficiency ρ = 1, cell temperature TC = 300 K. The above integrals converge only when Eg > eV, which is also consistent with the assumption that ρ = 1. For Eg-eV. kBTC, equation (I.3.39) can yield Voc higher than Eg/e,. which is obviously incorrect for Boltzmann statistics since the electronic populations are degenerate. Key assumptions are no longer valid; stimulated emission can take place in real material. Estimation of the Shockley-Queisser efficiency limit as a function of band gap energy Eg and applied voltage V is shown in figure I.14.. I.45 | P a g e.

(57) # The photovoltaic effect, the direct generation of electric power by light in a solid material, was discovered by British scientists: W. G. Adams and his student R. E. Day in the 1870s using selenium. An important breakthrough was made in the 1950s by G. Pearson, D. Chapin, and C. Fuller at Bell Labs. Using silicon, they demonstrated a solar cell of efficiency 5.7%, ten times greater than that of the selenium solar cell. Solar cells first found applications in space. The efficiency of silicon cells has been improved to about 24% in the early 2000s, very close to the theoretical limit of 28%. To date, semiconductor solar cells account for roughly 90% of the market share. Silicon solar cells account for more than 85% of the solar cell market. Thin film solar cells, especially those based on CIGS (copperದindiumದgalliumದ selenide) and CdTe-CdS, are second to silicon solar cells in market share.. #. !. $ %. When a p-type semiconductor and an n-type semiconductor are brought together, a built-in potential is established. Because the Fermi level of a p-type semiconductor is close to the top of the valence band and the Fermi-level of an n-type semiconductor is close to the bottom of the conduction band, there is a difference between the Fermi levels of the two sides. When the two pieces are combined to form a single system, the Fermi levels must be aligned. As a result, the energy levels of the two sides must undergo a shift with a potential V0. Letting Ecp be the energy level of the bottom of the conduction band for the p-type semiconductor versus the Fermi level and Ecn that for the n-type semiconductor, the built-in potential is:. w-. TU. TU*. (I.4.1). Because concentration of holes is low in the n-region, the holes diffuse from the p-region to the n-region (so called Fermi pressure). After a number of holes move to the n-region, an electrical field is formed to drive the holes back to the p-region. At equilibrium, the net current Jp(x) must be zero:. Í. w “æ. I.46 | P a g e. TQ. ç. ”. $. (I.4.2).

(58) Figure I.15. Formation of a p-n junction. When two pieces of semiconductor are brought together, the Fermi level must align. To achieve this, the holes in the p-side move to the nside, and electrons on n-side move to the p-side thus forming a double charged layer, until a dynamic equilibrium is established. where µp is the mobility of the holes, p(x) is the concentration of holes as a function of x, Ex(x) is the x-component of electric field intensity as a function of x, and Dp is the diffusion coefficient of the holes. Using Einstein's relation:. ç æ. Š. w. (I.4.3). and the relation between the potential V (x) and electric field intensity, Ex(x) = −dV(x)/dx, Eq. (I.4.2) becomes: w. Š. -. (I.4.4). Integrating Eq. (I.4.4) over the entire transition region yields: w. Š. œ-*. -•. xÄ. *. (I.4.5). Since Vn-Vp = V0, therefore we obtain: *. R. wŠ. S. (I.4.6). and a similar expression for electrons:. I.47 | P a g e.

(59) R. *. wŠ. S. (I.4.7). A very effective and fairly accurate model is based on the depletion approximation. Under such an approximation, in the S-region near the junction boundary there is a layer of thickness xp where all the holes are removed and the charge density ρp is determined by the density of the acceptors NA which are negatively charged: Ü. wk±. (I.4.8). The electrostatic potential Φin this region is given by the Poisson's equation: ¥. TQ. wk±. }. (I.4.9). Similarly, there is a slab of thickness xn where all the free electrons are removed, and the charge density ρn is determined by the density of the donors, ND, which are positively charged:. Ü*. wk-. (I.4.10). gives: ¥ É. }. TQ. wk-. (I.4.11). The boundary conditions for the p-n junction are as follows; the charge neutrality of the entire transition region requires that:. k-. k±. *. (I.4.12). Second, outside the transition region, the electric field should be zero:. TQ. $,èYe, Ò. ,[. , P. *. (I.4.13). Third, the electrostatic potential should match the values at the boundaries of the transition region:. ¥ ¥. $ [Ý,. - [Ý,. *. (I.4.14). The solution to equations (I.4.9), (I.4.11) with boundary conditions (I.4.14) is as follows:. I.48 | P a g e.

(60) wk± œ W }. TQ. wk± }. TQ. • èYe. Ò. èYe,$ ß. *. Ò. ß$ *. (I.4.15). Using the above boundary conditions and the definition of the width of the transition region W = xn+xp the following relation is obtained: -. w k± k· "} k± W k-. (I.4.16). ^. (I.4.17). and from it the transition region W as a function of V0 can be estimated: ·. "} R W S w k± k-. and the junction capacitance: } N ·. (I.4.18). The properties and carrier concentration of a p-n junction change when a bias voltage V is applied. By applying a forward bias, the potential difference across a p-n junction becomes V0-V. The electron concentration in the p-region, np, changes: |. •. *. w -. Š. -. ‘. (I.4.19). Comparing with equation (I.4.6), one finds anexcess free-electron concentration at the border of the neutral p-region: r. $. Î. R. wŠ. S. Ï. (I.4.20). Similarly, the external forward bias voltage V generates an excess hole concentration at the border of the neutral n-region: r. *. $. *. Î. R. wŠ. S. Ï. (I.4.21). The excess carrier concentrations generate an excess diffusion current, which is the main part of the forward-bias current of a diode.. I.49 | P a g e.

(61) Diffusion of excess minority carriers is the origin of junction current. However, there is a competing process which limits the junction current. The excess minority carriers are surrounded by a sea of majority carriers which are constantly courting for recombination. Because the concentration of majority carriers, pp or nn, is several orders of magnitude greater than the concentration of excess minority carriers, even with recombination, pp or nn is virtually a constant. The rate of decay of excess minority carriers is thus proportional to its concentration, which can be characterized by a lifetime. The combined effect of diffusion and lifetime of the excess minority carriers can be summarized in the following equations. For free electrons:. Ôr. ÔÝ. Ý. r. Ý. é*. W ç*. Ô r. Ý. Ô. (I.4.22). where Dn is the diffusion coefficient, and τn is the lifetime of free electrons. For holes:. Ôr. *. ÔÝ. Ý. r. Ý. *. é. Wç. Ô r. Ý. *. Ô. (I.4.23). where Dp is the diffusion coefficient and τp is the lifetime of holes. At equilibrium, the concentration of carriers is independent of time. For example equation (I.4.22) becomes: r. ç*. r. é*. (I.4.24). and the calculated diffusion current of electrons is: .*. wç*. r. ç* w^ r é*. (I.4.25). At x = 0 the junction current of electrons is: .* œ. $•. and for holes: .. *. $. ç* w^ é*. ç w^ é. *. Î Î. R R. wŠ. The total junction current is then:. I.50 | P a g e. wŠ. S. S. Ï Ï. (I.4.26). (I.4.27).

(62) .. w ¢^. ç* é*. W^. ç é. *¤ Î. R. wŠ. S. Ï. (I.4.28). Furthermore, using the approximate relations: h. h. k-. *. k±. (I.4.29). Equation (I.4.28) can be also expressed as follows: .. w. ç ç* ¢ ^ W ^ ¤Î k± é* k- é. h. R. wŠ. S. Ï. (I.4.30). Denoting a constant: .. w. h. ç ç* ¢ ^ W ^ ¤ k± é* k- é. (I.4.31). equation (I.4.30) is simplified to the well-known form of the diode equation, also known as the Shockley equation: .. . R. R. wŠ. S. S. (I.4.32). It is also worth noting that the minimum value of dark current I0 is given by: .. #. w. Š. !:. ). Q. T~ Š. (I.4.32). !. A solar cell is an electronic device which converts sunlight directly into electricity. Light shining on the solar cell produces both a current and a voltage to generate electric power. This process requires a material in which the absorption of light raises an electron to a higher energy state and allows the movement of this higher energy electron from the solar cell into an external circuit. Typical silicon solar cells are constructed as follows: the base is a piece of pದtype silicon, a fraction of a millimeter thick, lightly doped with boron. The emitter is. I.51 | P a g e.